Hsymmetric positions
These are the positions that are invariant under all symmetries
producing an even permutation of long diagonals of the cube.
There are 24 such positions; 4 have more symmetry, namely
Msymmetry.
The other 20 positions pair up into 10 patterns, as shown below.
I have even calculated all minimal maneuvers, using my
optimal cube solver.

Superfliptwist


U R F' B U' D' F U' D F L F' L' U R D F U R L

(20q*, 20f*)



Herbert
Kociemba [1, 3] was the first to find a 20f maneuver;
(it's different from this one).



Supertwist


U R' B D B U L D B' D^{2} R U' F L F R D L
F' L^{2}

(22q*, 20q)

B' L^{2} U D R^{2} B' D^{2} F^{2}
D' R^{2} F B L^{2} D' B^{2} U^{2}

(16f*, 24q)



Dik Winter [4] was the first to find a 16f maneuver; it's different
from this one.



6 H's


D^{2} L^{2} B^{2} U^{2}
D^{2} B^{2} R^{2} D^{2}

(16q*, 8f*)



Up to Hsymmetry, this is the only maneuver that is minimal in face
turns. (It is equivalent to its inverse.)
James Saxe [2] was the first to find this maneuver.



Superflip composed with 6 H's


U F' L' F' B U R F' B U' B' U D' R^{2} L' B U'

(18q*, 17f*)



Up to symmetry and inversion, this is the only maneuver that is
minimal in either quarter turns or face turns.



Superfliptwist composed with Pons Asinorum


U F B D R L U' F^{2} B^{2} R L D' F B D
R L D F' B'

(22q*, 20f*)



Up to symmetry and inversion, this is the only maneuver that is
minimal in quarter turns. It has the remarkable property that it
is invariant under a 180 degree rotation about the UD axis!
Dik Winter [5] was the first to find a 20f maneuver (different
from this one).



Supertwist composed with Pons Asinorum


F L D F U' B^{2} R F R' F' R F L^{2} U' R
D B R

(20q*, 18f)

B^{2} L U^{2} F' B' U^{2} R' F^{2}
L^{2} F' U^{2} R' L' U^{2} B R^{2}

(16f*, 24q)



The 20q maneuver is the unique minimal maneuver in quarter turns, up
to symmetry. (It is equivalent to its inverse.)
The 16f maneuver is the unique minimal maneuver in face turns, up
to symmetry and inversion.



Superfliptwist composed with 6 H's (type 1)


U F B U' R L U F B R^{2} L^{2} D' F B U'
R L D' R' L'

(22q*, 20f*)



This is the unique maneuver, up to symmetry, that is minimal in
quarter turns.
It has the remarkable property that it is invariant under a 180
degree rotation about the UD axis!



Superfliptwist composed with 6 H's (type 2)


Inverse of type 1



Supertwist composed with 6 H's (type 1)


U^{2} L U B D L U B' R' L' F' D R U F D L'
U^{2}

(20q*, 18f)

L' B^{2} U' D' B^{2} R' U^{2} L^{2}
U B^{2} R L F^{2} U R^{2} U^{2}

(16f*, 24q)




Supertwist composed with 6 H's (type 2)


Inverse of type 1


References
[1] Herbert Kociemba,
Close to God's Algorithm,
Cubism For Fun
28 (April 1992) pp. 1013.
[2] James Saxe,
16 qtw algs for CC and H patterns,
cubelovers email, December 16, 1980.
[3] Dik T. Winter,
Are we approaching God's algorithm?,
cubelovers email, May 4, 1992.
[4] Dik T. Winter,
Kociemba's algorithm, cubelovers email, May 17, 1992.
[5] Dik T. Winter,
Re: Diameter of cube group?, cubelovers email, August 25, 1993.
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Updated October 28, 2006.